Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

Cohen, Daniel E.; Madlener, Klaus; Otto, Friedrich

doi:10.1002/malq.19930390117

Cited by 13 publications

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“…[19]). More generally, in [35] it was shown that the isoperimetric function pa of a finitely presented subgroup G of a finitely presented group H is not recursively related with the isoperimetric function pn of H (see also [6,14], and Theorem 8.1 later in the present paper).…”

Section: Definitions and Basic Factsmentioning

confidence: 90%

“…In particular they prove that the embedding is necessary in the Theorem; in fact, a group G could have a fairly small computational complexity (and hence be embeddable in a group with a fairly small derivation distance), yet the derivation distance of G could be huge; see also [14]. A semigroup version of the necessity of an embedding appears in [6]; see also Theorem 8.1 in the present paper.…”

Section: Introductionmentioning

confidence: 91%

“…See also [14]. A simpler proof for finitely presented semigroups (with the Church-Rosser + Noetherian property, in addition) appears in [6,Theorem 3.4].…”

Section: The Embeddingmentioning

confidence: 97%

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Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

Birget

1998

Int. J. Algebra Comput.

147

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The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem:Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) > T(n)+T(m)).Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is 0(T(|a;| + \v\) 2 )-Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(T 0 )) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(T D )).In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function < D (where D(n) > n), then the word problem of S has nondeterministic time complexity 0(D).The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T)°' 1 ')) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) 0 ' 1 ') isoperimetric function.An algorithmic problem L is in NP (or more generally, in NTIME^T) 0^) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) 0 ' 1 ') isoperimetric function.In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

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Section: Definitions and Basic Factsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 91%

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Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

Birget

1998

Int. J. Algebra Comput.

147

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“…Note that by Proposition 2.20, the space complexity of the word problem in a finitely presented group G does not exceed the FFFL function of G. It follows from [53,150] that the converse statement fails. An easier example is given by the Baumslag 1-relator group from [14] G = a, b | (aba −1 )b(aba −1 ) −1 = b 2 .…”

Section: The Space Functions Of Turing Machines and The Filling Functmentioning

confidence: 73%

Asymptotic invariants, complexity of groups and related problems

Sapir

2011

Bull. Math. Sci.

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“…Cohen, Madlener & Otto built the first examples. in a series of papers [7,8,26] where Dehn functions were first defined. They designed their groups in such a way that the 'intrinsic' method of solving the word problem involves running a very slow algorithm which has been suitably 'embedded' in the presentation.…”

mentioning

confidence: 99%

Taming the hydra: The word problem and extreme integer compression

Dison

Einstein

Riley

2018

Int. J. Algebra Comput.

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For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison & Riley showed that a "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.and so it is possible to check efficiently when a word on a ±1 and b ±1 represents the identity by multiplying out the corresponding string of matrices.A celebrated 1-relator group due to Baumslag [1] provides a more dramatic example:Platonov [29] proved its Dehn function grows like n → ⌊log 2 n⌋ exp 2 ( exp 2 · · · (exp 2 (1)) · · · ), where exp 2 (n) := 2 n . (Earlier results in this direction are in [2,14,15].) Nevertheless, Miasnikov, Ushakov & Won [27] solve its word problem in polynomial time. (In unpublished work I. Kapovich and Schupp showed it is solvable in exponential time [33].)another example. Diekert, Laun & Ushakov [11] recently gave a polynomial time algorithm for its word problem and, citing a 2010 lecture of Bridson, claim it too has Dehn function growing like a tower of exponentials. The groups we focus on in this article are yet more extreme 'natural examples'. They arose in the study of hydra groups by Dison & Riley [12] . Let θ : F(a 1 , . . . , a k ) → F(a 1 , . . . , a k )be the automorphism of the free group of rank k such that θ(a 1 ) = a 1 and θ(a i ) = a i a i−1 for i = 2, . . . , k. The family G k := a 1 , . . . , a k , t | t −1 a i t = θ(a i ) ∀i > 1 , are called hydra groups. Take HNN-extensions Γ k := a 1 , . . . , a k , t, p | t −1 a i t = θ(a i ), [p, a i t] = 1 ∀i > 1 TAMING THE HYDRA 5 of G k where the stable letter p commutes with all elements of the subgroup H k := a 1 t, . . . , a k t .It is shown in [12] that for k = 1, 2, . . ., the subgroup H k is free of rank k and Γ k has Dehn function growing like n → A k (n). Here we prove that nevertheless:Theorem 2. For all k, the word problem of Γ k is solvable in polynomial time.(In fact, our algorithm halts within time bounded above by a polynomial of degree 3k 2 + k + 2-see Section 5.)1.3. The membership problem and subgroup distortion. Distortion is the root cause of the Dehn function of Γ k growing like n → A k (n). The massive gap between Dehn function and the time-complexity of the word problem for Γ k is attributable to a similarly massive gap between a distortion function and the time-complexity of a membership problem. Here are more details.Suppose H is a subgroup of a group G and G and H have finite generating sets S and T , respectively. So G has a word metric d S (g, h), the length of a shortest word on S ±1 representing g −1 h, and H has a word metric d T s...

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Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

Cited by 13 publications

References 7 publications

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

Asymptotic invariants, complexity of groups and related problems

Taming the hydra: The word problem and extreme integer compression

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